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<h1>Figure 6.2: Penalty function approximation</h1>
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<pre class="codeinput">
<span class="comment">% Section 6.1.2</span>
<span class="comment">% Boyd &amp; Vandenberghe "Convex Optimization"</span>
<span class="comment">% Original by Lieven Vandenberghe</span>
<span class="comment">% Adapted for CVX Argyris Zymnis - 10/2005</span>
<span class="comment">%</span>
<span class="comment">% Comparison of the ell1, ell2, deadzone-linear and log-barrier</span>
<span class="comment">% penalty functions for the approximation problem:</span>
<span class="comment">%       minimize phi(A*x-b),</span>
<span class="comment">%</span>
<span class="comment">% where phi(x) is the penalty function</span>
<span class="comment">% Log-barrier will be implemented in the future version of CVX</span>

<span class="comment">% Generate input data</span>
randn(<span class="string">'state'</span>,0);
m=100; n=30;
A = randn(m,n);
b = randn(m,1);

<span class="comment">% ell_1 approximation</span>
<span class="comment">% minimize   ||Ax+b||_1</span>
disp(<span class="string">'ell-one approximation'</span>);
cvx_begin
    variable <span class="string">x1(n)</span>
    minimize(norm(A*x1+b,1))
cvx_end

<span class="comment">% ell_2 approximation</span>
<span class="comment">% minimize ||Ax+b||_2</span>
disp(<span class="string">'ell-2'</span>);
x2=-A\b;

<span class="comment">% deadzone penalty approximation</span>
<span class="comment">% minimize sum(deadzone(Ax+b,0.5))</span>
<span class="comment">% deadzone(y,z) = max(abs(y)-z,0)</span>
dz = 0.5;
disp(<span class="string">'deadzone penalty'</span>);
cvx_begin
    variable <span class="string">xdz(n)</span>
    minimize(sum(max(abs(A*xdz+b)-dz,0)))
cvx_end


<span class="comment">% log-barrier penalty approximation</span>
<span class="comment">%</span>
<span class="comment">% minimize -sum log(1-(ai'*x+bi)^2)</span>

disp(<span class="string">'log-barrier'</span>)

<span class="comment">% parameters for Newton Method &amp; line search</span>
alpha=.01; beta=.5;

<span class="comment">% minimize linfty norm to get starting point</span>
cvx_begin
    variable <span class="string">xlb(n)</span>
    minimize <span class="string">norm(A*xlb+b,Inf)</span>
cvx_end
linf = cvx_optval;
A = A/(1.1*linf);
b = b/(1.1*linf);

<span class="keyword">for</span> iters = 1:50

   yp = 1 - (A*xlb+b);  ym = (A*xlb+b) + 1;
   f = -sum(log(yp)) - sum(log(ym));
   g = A'*(1./yp) - A'*(1./ym);
   H = A'*diag(1./(yp.^2) + 1./(ym.^2))*A;
   v = -H\g;
   fprime = g'*v;
   ntdecr = sqrt(-fprime);
   <span class="keyword">if</span> (ntdecr &lt; 1e-5), <span class="keyword">break</span>; <span class="keyword">end</span>;

   t = 1;
   newx = xlb + t*v;
   <span class="keyword">while</span> ((min(1-(A*newx +b)) &lt; 0) | (min((A*newx +b)+1) &lt; 0))
       t = beta*t;
       newx = xlb + t*v;
   <span class="keyword">end</span>;
   newf = -sum(log(1 - (A*newx+b))) - sum(log(1+(A*newx+b)));
   <span class="keyword">while</span> (newf &gt; f + alpha*t*fprime)
       t = beta*t;
       newx = xlb + t*v;
       newf = -sum(log(1-(A*newx+b))) - sum(log(1+(A*newx+b)));
   <span class="keyword">end</span>;
   xlb = xlb+t*v;
<span class="keyword">end</span>


<span class="comment">% Plot histogram of residuals</span>

ss = max(abs([A*x1+b; A*x2+b; A*xdz+b;  A*xlb+b]));
tt = -ceil(ss):0.05:ceil(ss);  <span class="comment">% sets center for each bin</span>
[N1,hist1] = hist(A*x1+b,tt);
[N2,hist2] = hist(A*x2+b,tt);
[N3,hist3] = hist(A*xdz+b,tt);
[N4,hist4] = hist(A*xlb+b,tt);


range_max=2.0;  rr=-range_max:1e-2:range_max;

figure(1), clf, hold <span class="string">off</span>
subplot(4,1,1),
bar(hist1,N1);
hold <span class="string">on</span>
plot(rr, abs(rr)*40/3, <span class="string">'-'</span>);
ylabel(<span class="string">'p=1'</span>)
axis([-range_max range_max 0 40]);
hold <span class="string">off</span>

subplot(4,1,2),
bar(hist2,N2);
hold <span class="string">on</span>;
plot(rr,2*rr.^2),
ylabel(<span class="string">'p=2'</span>)
axis([-range_max range_max 0 11]);
hold <span class="string">off</span>

subplot(4,1,3),
bar(hist3,N3);
hold <span class="string">on</span>
plot(rr,30/3*max(0,abs(rr)-dz))
ylabel(<span class="string">'Deadzone'</span>)
axis([-range_max range_max 0 25]);
hold <span class="string">off</span>

subplot(4,1,4),
bar(hist4,N4);
rr_lb=linspace(-1+(1e-6),1-(1e-6),600);
hold <span class="string">on</span>
plot(rr_lb, -3*log(1-rr_lb.^2),rr,2*rr.^2,<span class="string">'--'</span>)
axis([-range_max range_max 0 11]);
ylabel(<span class="string">'Log barrier'</span>),
xlabel(<span class="string">'r'</span>)
hold <span class="string">off</span>
</pre>
<a id="output"></a>
<pre class="codeoutput">
ell-one approximation
 
Calling SDPT3 4.0: 230 variables, 100 equality constraints
------------------------------------------------------------

 num. of constraints = 100
 dim. of socp   var  = 200,   num. of socp blk  = 100
 dim. of free   var  = 30 *** convert ublk to lblk
*******************************************************************
   SDPT3: Infeasible path-following algorithms
*******************************************************************
 version  predcorr  gam  expon  scale_data
    NT      1      0.000   1        0    
it pstep dstep pinfeas dinfeas  gap      prim-obj      dual-obj    cputime
-------------------------------------------------------------------
 0|0.000|0.000|9.0e-01|6.3e+01|3.2e+05| 2.091344e+02  0.000000e+00| 0:0:00| chol  1  1 
 1|1.000|0.882|1.0e-05|7.5e+00|2.0e+04| 5.193045e+03  5.169955e+01| 0:0:00| chol  1  1 
 2|1.000|0.990|4.3e-06|9.5e-02|1.6e+03| 1.578557e+03  2.478944e+01| 0:0:00| chol  1  1 
 3|0.953|1.000|2.3e-06|3.0e-03|7.5e+01| 1.031902e+02  2.866779e+01| 0:0:00| chol  1  1 
 4|0.784|0.271|3.6e-05|2.3e-03|3.3e+01| 6.804505e+01  3.545933e+01| 0:0:00| chol  1  1 
 5|0.920|0.209|1.8e-05|1.8e-03|2.1e+01| 5.972104e+01  3.951007e+01| 0:0:00| chol  1  1 
 6|0.928|0.439|8.9e-06|1.0e-03|1.1e+01| 5.698499e+01  4.626798e+01| 0:0:00| chol  1  1 
 7|0.975|0.470|4.8e-06|5.4e-04|5.4e+00| 5.564218e+01  5.036971e+01| 0:0:00| chol  1  1 
 8|1.000|0.328|3.5e-07|3.6e-04|3.6e+00| 5.542723e+01  5.188850e+01| 0:0:00| chol  1  1 
 9|1.000|0.474|5.0e-08|1.9e-04|1.9e+00| 5.523572e+01  5.340311e+01| 0:0:00| chol  1  1 
10|1.000|0.617|1.8e-08|7.3e-05|7.0e-01| 5.515308e+01  5.446389e+01| 0:0:00| chol  1  1 
11|0.712|0.265|1.9e-08|5.4e-05|5.1e-01| 5.514646e+01  5.463760e+01| 0:0:00| chol  1  1 
12|1.000|0.216|1.2e-08|4.2e-05|4.1e-01| 5.514672e+01  5.474159e+01| 0:0:00| chol  1  1 
13|1.000|0.448|6.2e-09|2.3e-05|2.2e-01| 5.513610e+01  5.491355e+01| 0:0:00| chol  1  1 
14|1.000|0.780|1.3e-09|5.1e-06|4.9e-02| 5.512960e+01  5.508124e+01| 0:0:00| chol  1  1 
15|0.986|0.975|1.5e-10|1.3e-07|1.2e-03| 5.512893e+01  5.512772e+01| 0:0:00| chol  1  1 
16|0.989|0.989|1.4e-11|1.6e-06|2.3e-05| 5.512892e+01  5.512891e+01| 0:0:00| chol  1  1 
17|0.973|0.945|3.9e-13|3.1e-08|1.0e-06| 5.512892e+01  5.512892e+01| 0:0:00| chol  1  1 
18|0.569|0.944|1.7e-13|1.4e-09|1.0e-07| 5.512892e+01  5.512892e+01| 0:0:00|
  stop: max(relative gap, infeasibilities) &lt; 1.49e-08
-------------------------------------------------------------------
 number of iterations   = 18
 primal objective value =  5.51289216e+01
 dual   objective value =  5.51289215e+01
 gap := trace(XZ)       = 1.00e-07
 relative gap           = 9.00e-10
 actual relative gap    = 8.41e-10
 rel. primal infeas (scaled problem)   = 1.70e-13
 rel. dual     "        "       "      = 1.39e-09
 rel. primal infeas (unscaled problem) = 0.00e+00
 rel. dual     "        "       "      = 0.00e+00
 norm(X), norm(y), norm(Z) = 1.2e+01, 8.8e+00, 1.3e+01
 norm(A), norm(b), norm(C) = 7.9e+01, 1.0e+01, 1.1e+01
 Total CPU time (secs)  = 0.26  
 CPU time per iteration = 0.01  
 termination code       =  0
 DIMACS: 5.8e-13  0.0e+00  7.6e-09  0.0e+00  8.4e-10  9.0e-10
-------------------------------------------------------------------
 
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +55.1289
 
ell-2
deadzone penalty
 
Calling SDPT3 4.0: 430 variables, 200 equality constraints
------------------------------------------------------------

 num. of constraints = 200
 dim. of socp   var  = 200,   num. of socp blk  = 100
 dim. of linear var  = 200
 dim. of free   var  = 30 *** convert ublk to lblk
*******************************************************************
   SDPT3: Infeasible path-following algorithms
*******************************************************************
 version  predcorr  gam  expon  scale_data
    NT      1      0.000   1        0    
it pstep dstep pinfeas dinfeas  gap      prim-obj      dual-obj    cputime
-------------------------------------------------------------------
 0|0.000|0.000|1.6e+00|6.5e+01|6.0e+05| 4.182689e+03  0.000000e+00| 0:0:00| chol  1  1 
 1|0.827|0.379|2.8e-01|4.0e+01|1.8e+05| 1.111517e+04 -4.129673e+02| 0:0:00| chol  1  1 
 2|1.000|0.985|2.9e-05|6.6e-01|1.1e+04| 8.973038e+03 -5.186638e+01| 0:0:00| chol  1  1 
 3|1.000|1.000|1.7e-06|2.7e-02|3.0e+02| 2.473161e+02 -4.390984e+01| 0:0:00| chol  1  1 
 4|0.914|0.627|1.2e-05|1.2e-02|7.9e+01| 5.428420e+01 -2.255807e+01| 0:0:00| chol  1  1 
 5|0.988|0.248|1.3e-06|8.9e-03|5.4e+01| 3.890804e+01 -1.312735e+01| 0:0:00| chol  1  1 
 6|1.000|0.392|1.5e-06|5.4e-03|3.7e+01| 3.678911e+01  1.030166e+00| 0:0:00| chol  1  1 
 7|1.000|0.429|1.7e-07|3.1e-03|1.7e+01| 2.593389e+01  9.083688e+00| 0:0:00| chol  1  1 
 8|0.783|0.412|7.8e-08|1.8e-03|1.0e+01| 2.407722e+01  1.378554e+01| 0:0:00| chol  1  1 
 9|1.000|0.279|1.6e-08|1.3e-03|7.2e+00| 2.276879e+01  1.572590e+01| 0:0:00| chol  1  1 
10|1.000|0.408|1.4e-08|7.8e-04|4.4e+00| 2.219874e+01  1.788415e+01| 0:0:00| chol  1  1 
11|1.000|0.471|5.5e-09|4.1e-04|2.3e+00| 2.170847e+01  1.949236e+01| 0:0:00| chol  1  1 
12|1.000|0.301|1.1e-09|2.9e-04|1.6e+00| 2.164468e+01  2.004788e+01| 0:0:00| chol  1  1 
13|1.000|0.529|9.4e-10|1.4e-04|7.5e-01| 2.152118e+01  2.077856e+01| 0:0:00| chol  1  1 
14|1.000|0.414|4.2e-10|7.9e-05|4.4e-01| 2.149022e+01  2.105693e+01| 0:0:00| chol  1  1 
15|1.000|0.672|8.3e-11|2.6e-05|1.4e-01| 2.147157e+01  2.133151e+01| 0:0:00| chol  1  1 
16|1.000|0.876|9.4e-12|3.2e-06|1.7e-02| 2.146833e+01  2.145121e+01| 0:0:00| chol  1  1 
17|0.988|0.984|2.7e-12|1.3e-05|3.5e-04| 2.146821e+01  2.146794e+01| 0:0:00| chol  1  1 
18|1.000|0.989|3.9e-15|2.7e-07|6.9e-06| 2.146821e+01  2.146821e+01| 0:0:00| chol  1  1 
19|1.000|0.989|2.7e-13|5.3e-09|1.2e-07| 2.146821e+01  2.146821e+01| 0:0:00|
  stop: max(relative gap, infeasibilities) &lt; 1.49e-08
-------------------------------------------------------------------
 number of iterations   = 19
 primal objective value =  2.14682118e+01
 dual   objective value =  2.14682117e+01
 gap := trace(XZ)       = 1.19e-07
 relative gap           = 2.71e-09
 actual relative gap    = 2.17e-09
 rel. primal infeas (scaled problem)   = 2.74e-13
 rel. dual     "        "       "      = 5.26e-09
 rel. primal infeas (unscaled problem) = 0.00e+00
 rel. dual     "        "       "      = 0.00e+00
 norm(X), norm(y), norm(Z) = 1.2e+01, 9.5e+00, 1.3e+01
 norm(A), norm(b), norm(C) = 8.1e+01, 1.1e+01, 1.1e+01
 Total CPU time (secs)  = 0.33  
 CPU time per iteration = 0.02  
 termination code       =  0
 DIMACS: 1.1e-12  0.0e+00  2.9e-08  0.0e+00  2.2e-09  2.7e-09
-------------------------------------------------------------------
 
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +21.4682
 
log-barrier
 
Calling SDPT3 4.0: 300 variables, 131 equality constraints
   For improved efficiency, SDPT3 is solving the dual problem.
------------------------------------------------------------

 num. of constraints = 131
 dim. of socp   var  = 200,   num. of socp blk  = 100
 dim. of linear var  = 100
*******************************************************************
   SDPT3: Infeasible path-following algorithms
*******************************************************************
 version  predcorr  gam  expon  scale_data
    NT      1      0.000   1        0    
it pstep dstep pinfeas dinfeas  gap      prim-obj      dual-obj    cputime
-------------------------------------------------------------------
 0|0.000|0.000|1.2e+02|1.6e+01|2.2e+04| 0.000000e+00  0.000000e+00| 0:0:00| chol  1  1 
 1|0.978|1.000|2.5e+00|1.0e-01|5.0e+02|-8.798474e-02 -2.283362e+01| 0:0:00| chol  1  1 
 2|1.000|1.000|1.1e-07|1.0e-02|1.7e+01|-8.440836e-02 -1.693198e+01| 0:0:00| chol  1  1 
 3|1.000|0.851|4.4e-08|2.3e-03|2.8e+00|-2.514918e-01 -3.057397e+00| 0:0:00| chol  1  1 
 4|0.983|0.888|1.3e-08|3.5e-04|1.0e+00|-7.769010e-01 -1.813573e+00| 0:0:00| chol  1  1 
 5|0.781|0.851|2.8e-09|6.1e-05|4.1e-01|-9.903560e-01 -1.401445e+00| 0:0:00| chol  1  1 
 6|0.827|0.988|5.6e-10|1.7e-06|1.6e-01|-1.100530e+00 -1.264575e+00| 0:0:00| chol  1  1 
 7|0.789|0.754|3.5e-10|5.0e-07|6.4e-02|-1.161363e+00 -1.225015e+00| 0:0:00| chol  1  1 
 8|0.880|0.922|5.2e-11|4.9e-08|1.9e-02|-1.187790e+00 -1.207002e+00| 0:0:00| chol  1  1 
 9|0.678|0.960|1.4e-11|2.9e-09|5.4e-03|-1.196463e+00 -1.201855e+00| 0:0:00| chol  1  1 
10|0.872|1.000|1.8e-12|1.0e-10|1.6e-03|-1.199955e+00 -1.201547e+00| 0:0:00| chol  1  1 
11|0.946|0.967|9.5e-14|1.4e-11|8.3e-05|-1.201197e+00 -1.201280e+00| 0:0:00| chol  1  1 
12|0.984|0.808|1.1e-14|3.7e-12|5.9e-06|-1.201267e+00 -1.201273e+00| 0:0:00| chol  1  1 
13|0.880|1.000|6.7e-12|1.0e-12|9.8e-07|-1.201270e+00 -1.201271e+00| 0:0:00| chol  1  1 
14|0.638|1.000|9.3e-12|1.3e-12|5.0e-07|-1.201270e+00 -1.201271e+00| 0:0:00| chol  1  1 
15|0.643|1.000|3.7e-12|1.9e-12|2.7e-07|-1.201270e+00 -1.201271e+00| 0:0:00| chol  1  1 
16|0.644|1.000|1.5e-12|1.0e-12|1.4e-07|-1.201270e+00 -1.201270e+00| 0:0:00| chol  1  1 
17|0.645|1.000|5.7e-13|1.0e-12|7.4e-08|-1.201270e+00 -1.201270e+00| 0:0:00| chol  1  1 
18|0.645|1.000|2.2e-13|1.0e-12|3.9e-08|-1.201270e+00 -1.201270e+00| 0:0:00|
  stop: max(relative gap, infeasibilities) &lt; 1.49e-08
-------------------------------------------------------------------
 number of iterations   = 18
 primal objective value = -1.20127043e+00
 dual   objective value = -1.20127047e+00
 gap := trace(XZ)       = 3.94e-08
 relative gap           = 1.16e-08
 actual relative gap    = 1.16e-08
 rel. primal infeas (scaled problem)   = 2.18e-13
 rel. dual     "        "       "      = 1.00e-12
 rel. primal infeas (unscaled problem) = 0.00e+00
 rel. dual     "        "       "      = 0.00e+00
 norm(X), norm(y), norm(Z) = 3.8e-01, 3.5e+00, 1.4e+01
 norm(A), norm(b), norm(C) = 5.8e+01, 2.0e+00, 1.0e+01
 Total CPU time (secs)  = 0.24  
 CPU time per iteration = 0.01  
 termination code       =  0
 DIMACS: 2.2e-13  0.0e+00  3.4e-12  0.0e+00  1.2e-08  1.2e-08
-------------------------------------------------------------------
 
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +1.20127
 
</pre>
<a id="plots"></a>
<div id="plotoutput">
<img src="penalty_comp_cvx__01.png" alt=""> 
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